Since \(\Pi \)  is PM-compatible, \(\rhd _{\Pi (A)}\)  is a total preorder. Therefore there are no `cycles' and by Theorem REF  we have that there exists at least one alternative \(a\in A\)  that PM-dominates the other alternatives. According to the results in Fishburn [1]} and Felsenthal and Tideman [2]}, we know that Black, Dodgson, Young, Minimax and Fishburn under the absence of cycles are Strict Condorcet Consistent. That is, they select the set of candidates that beat or tie with all other alternatives.The unbeaten alternatives are referred to in the literature as Weak Condorcet winners or Quasi-Condorcet candidates. It is also clear that the Schwartz rule is Weak Condorcet Consistent. Therefore we have that \(f(\Pi (A))=\lbrace a\in A:a\rhd _{\Pi }^{pm}b\, for\, all\, b\in A\rbrace \)  for these rules. It is straightforward from the definition of Weak Condorcet winners that these rules are strongly PM-efficient.
